LONDON—Saudi Arabia’s potential sale of shares in its state-owned oil giant could lead to a publicly listed company valued in the trillions of dollars, more than 10 times Apple Inc.’s peak of about $756 billion.

Saudi Arabian Oil Co., better known as Saudi Aramco, on Friday held out the prospect of an IPO on the Saudi stock exchange. Aramco said it was considering “the listing in capital markets of an appropriate percentage of the company’s shares and/or the listing of a bundle of its downstream subsidiaries.”